An Analytic-Numerical Method for Computation of the Liapunov and Period Constants Derived from Their Algebraic Structure

Abstract

Abstract. We consider the problem of computing the Liapunov and the period constants for a smooth di erential equation with a non degenerate critical point. First, we i n vestigate the structure of both constants when they are regarded as polynomials on the coe cients of the di erential equation. Secondly, w e t a k e advantadge of this structure to derive a method to obtain the explicit expression of the above-mentioned constants. Although this method is based on the use of the Runge-KuttaFehlberg methods of orders 7 and 8 and the use of Richardson's extrapolation, it provides the real expression for these constants.

New Advances on the Lyapunov Constants of Some Families of Planar Differential Systems

“…Observe that if all L k (λ) vanish thenḞ = 0, and therefore F is a first integral, so the origin is a center. These L k (λ) are actually the Lyapunov constants, which are polynomials in the parameters λ (see [4]). For the sake of simplicity, we will denote them simply as L k .…”

“…We define the quasi degree of M as k, (k + − 1)(p k + q k ) and its weight as k, (1 − k − )(p k − q k ). Then, by [4], the monomials of a Lyapunov constant L j satisfy that they have quasi-degree 2j and weight 0. Now using these properties together with the degree of L j , we can select which monomials are candidates to be part of each A j , but with undetermined coefficients.…”

Criticality via first order development of the period constants

“…As we have already mentioned, the studied polynomial systems have homogeneous nonlinearities of degree n. We will consider systems with even n, and the reason is as follows. It is a well-known fact that, given a parametric family of systems, its period constants are polynomials whose variables are the parameters of the system and having a particular structure based on their weight and quasi-degree (for more details see for instance [10,14]). It can be checked that this structure implies that, when the nonlinearities are homogeneous of degree n, some of the corresponding period constants are identically zero.…”

“…2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 2 , 18, 19 2 , 20 2 , 21, 22 2 , 23 9 ), for (35) and r 35 = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 2 , 18, 19 2 , 20 3 , 21 11 ),…”

On the Integrability of Persistent Quadratic Three-Dimensional Systems

“…Generally, we are not able to calculate a large number of Lyapunov constants due to the limitations of computers. We need to know in which step the calculation of Lyapunov constants can be stopped (in the sense that we have obtained enough quantities/polynomials) not only using computer technology but also using some other analytic and algebraic tools; see [4,5].…”